Ratios & Proportions
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SAT Math › Ratios & Proportions
During the same time a cyclist rides 15 miles, a runner runs 6 miles. If the runner's speed is 8 miles/hour, what is the cyclist's speed in miles/hour?
3.2
10
14
20
Explanation
Because time is the same, speeds are proportional to distances: 15/6 times 8 is 20 mph. The other choices come from inverting the ratio, adding distances to the speed, or averaging incorrectly.
In a club, the ratio of boys to girls is 5:7. After 4 boys leave and 6 girls join, the ratio becomes 2:3. How many students were originally in the club?
144
156
276
288
Explanation
Let boys = 5k and girls = 7k; solving $(5k - 4)/(7k + 6) = 2/3$ gives k = 24, so total originally = 12k = 288. The distractors result from misapplying the ratio or halving totals.
A runner goes up a hill at 6 km/h and returns down the same route at 12 km/h. The round-trip distance is 9 miles (use 1 mile ≈ 1.6 km). How many minutes does the round trip take?
72
96
108
135
Explanation
Convert 9 miles ≈ 14.4 km; times are 7.2/6 + 7.2/12 = 1.8 hours = 108 minutes. 96 uses a simple average speed, 135 uses miles without converting, and 72 uses the faster speed for the whole trip.
A 50-liter tank holds a solution that is 40% acid. Some liquid is removed and replaced with water, leaving the tank full; the new concentration is 28% acid. How many liters were removed?
6
12
15
36
Explanation
Acid after replacement is 0.4(50 − x) = 0.28(50) = 14, giving x = 15. 6 sets 0.4×50 − x = 14, 12 is a proportionality mistake, and 36 comes from misinterpreting percentages.
A drink mix is combined with water in the ratio 1:7 by volume. To fill a 5.6-liter container, how many milliliters of concentrate are needed?
70
700
800
4900
Explanation
Concentrate is 1/8 of the total, so 5.6 L × 1/8 = 0.7 L = 700 mL. 800 mL uses 1/7 instead of 1/(1+7), 4900 mL uses 7/8, and 70 mL mis-converts liters to milliliters.
Two meshed gears have 18 teeth (Gear A) and 30 teeth (Gear B). If Gear A spins at 1500 rpm, what is Gear B's angular speed in degrees per second?
900
5400
15000
324000
Explanation
Gear speeds are inversely proportional to teeth, so rpm_B = 1500 × 18/30 = 900 rpm, which is 900 × 360/60 = 5400 degrees/second. 15000 inverts the ratio, 324000 forgets to convert minutes to seconds, and 900 treats rpm as degrees/sec.
On a map, the scale is 1 cm : 25 km. A route measures 14 cm on the map, and a car travels at 60 miles/hour. Using 1 km ≈ 0.62 miles, about how many hours will the trip take?
3.6
4.3
5.8
9.3
Explanation
Distance is 14 × 25 = 350 km ≈ 350 × 0.62 = 217 miles, so time ≈ 217/60 ≈ 3.6 hours. 5.8 ignores unit conversion, 9.3 uses the inverse km-to-mile factor, and 4.3 uses the wrong speed.
A smoothie mix uses fruit to yogurt in a ratio 3:2. If 30 ounces of mix are made and then 20% is spilled before serving, how many ounces of yogurt are served?
6
9.6
12
14.4
Explanation
Yogurt is 2/5 of 30 = 12 ounces, and after a 20% spill only 80% remains: 12 × 0.8 = 9.6. 12 ignores the spill, 14.4 uses the fruit fraction 3/5, and 6 subtracts 20% of the total from the yogurt amount.
A cyclist travels 18 miles in 45 minutes at a constant speed. At this speed, how many minutes will it take to travel 30 miles?
27
54
72
75
Explanation
Time scales with distance: 30/18 of 45 minutes = 75 minutes. The other choices reflect inverting the proportion or mixing up multiplicative scaling.
A printer produces 12 pages in 30 seconds. At this rate, how many pages will it produce in 4 minutes?
24
48
60
96
Explanation
Rate = 12/30 = 0.4 pages/second, and 4 minutes is 240 seconds, so pages = 0.4×240 = 96. The other choices come from using minutes instead of seconds or inverting the rate.